Homework8.ipynb

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   "source": [
    "# Homework Set 8"
   ]
  },
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     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
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    "%pylab inline"
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    "## Problem 1:\n",
    "\n",
    "Derive the weak duality property by proving the following statement:\n",
    "\n",
    "* If $\\mathbf{x}^*$ is a solution to the original optimization problem (**Primal Problem**), then the Lagrange Dual function defined in the slides satisfies: for $\\forall \\mathbf{\\lambda} \\succeq 0$ and any $\\mathbf{\\mu}$,\n",
    "\n",
    "$$\n",
    "\\hat{f}(\\mathbf{\\mu}, \\mathbf{\\lambda}) \\leq f(\\mathbf{x}^*).\n",
    "$$"
   ]
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   "source": [
    "## Problem 2\n",
    "\n",
    "Derive the dual problem for the portfolio optimization problem:\n",
    "\n",
    "\n",
    "\\begin{aligned}\n",
    "\\min_{\\mathbf{x}} \\;\\; \\frac{1}{2} \\lambda\\; \\mathbf{x}^T \\Sigma \\mathbf{x} &- \\mu^T \\mathbf{x}\n",
    "\\\\\n",
    "s.t. \\;\\;\\sum_i x_i &= 1\n",
    "\\end{aligned}\n",
    "\n",
    "where $\\lambda$ is the risk-aversion coefficient, $\\mu$ is the expected asset return vector and $\\Sigma$ is the covariance matrix. \n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3\n",
    "\n",
    "Prove the statement that for convex problems, any local minimum is also a global minimum.\n",
    "\n",
    "Hint: proof by contradiction. Define the ball for the local minimum region, and assume there is a lower global minimum, find a point within the ball such that its function value is less than the local minimum value.\n"
   ]
  }
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